Noncommutativity from Canonical and Noncanonical Structures

TL;DR

This paper develops a formalism using symplectic structures and Dirac's quantization to handle theories with space-space and space-time noncommutativity, allowing for different quantizations based on the choice of potentials and transformations.

Contribution

It introduces a unified approach to quantize noncommutative theories via actions, accommodating both canonical and noncanonical structures, and relates different quantizations through transformations or Darboux maps.

Findings

Certain potentials lead to equivalent quantizations via canonical transformations.

Other potentials require Darboux maps, resulting in distinct quantum theories.

The formalism applies to both particle systems and field theories.

Abstract

Using arbitrary symplectic structures and parametrization invariant actions, we develop a formalism, based on Dirac's quantization procedure, that allows us to consider theories with both space-space as well as space-time noncommutativity. Because the formalism has as a starting point an action, the procedure admits quantizing the theory either by obtaining the quantum evolution equations or by using the path integral techniques. For both approaches we only need to select a complete basis of commutative observables. We show that for certain choices of the potentials that generate a given symplectic structure, the phase of the quantum transition function between the admissible bases corresponds to a linear canonical transformation, by means of which the actions associated to each of these bases may be related and hence lead to equivalent quantizations. There are however other potentials that result in actions which can not be related to the previous ones by canonical transformations, and for which the fixed end-points, in terms of the admissible bases, can only be realized by means of a Darboux map. In such cases the original arbitrary symplectic structure is reduced to its canonical form and therefore each of these actions results in a different quantum theory. One interesting feature of the formalism here discussed is that it can be introduced both at the levels of particle systems as well as of field theory.